Library GeoCoq.Meta_theory.Parallel_postulates.par_trans_playfair

Require Import GeoCoq.Axioms.parallel_postulates.
Require Import GeoCoq.Tarski_dev.Ch12_parallel.

Section par_trans_playfair.

Context `{T2D:Tarski_2D}.

Lemma par_trans_implies_playfair :
  postulate_of_transitivity_of_parallelism → playfair_s_postulate.
Proof.
intros HPT A1; intros.
assert (Par C1 C2 B1 B2) by (apply HPT with A1 A2; Par).
induction H3.
exfalso; apply H3; ∃ P; Col.
spliter; split; Col.
Qed.

End par_trans_playfair.